Bifurcation and control for a discrete-time prey–predator model with Holling-IV functional response

Abstract

The dynamics of a discrete-time predator-prey model with Holling-IV functional response are investigated. It is shown that the model undergoes a flip bifurcation, a Hopf bifurcation and a saddle-node bifurcation by using the center manifold theorem and bifurcation theory. Numerical simulations not only exhibit our results with the theoretical analysis, but also show the complex dynamical behaviors, such as the period-3, 6, 9, 12, 20, 63, 70, 112 orbits, a cascade of period-doubling bifurcations in period-2, 4, 8, 16, quasi-periodic orbits, an attracting invariant circle, an inverse period-doubling bifurcation from the period-32 orbit leading to chaos and a boundary crisis, a sudden onset of chaos and a sudden disappearance of the chaotic dynamics, attracting chaotic sets and non-attracting sets. We also observe that when the prey is in chaotic dynamics the predator can tend to extinction or to a stable equilibrium. Specifically, we stabilize the chaotic orbits at an unstable fixed point by using OGY chaotic control.

Permanence of a predator–prey discrete system with Holling-IV functional response and distributed delays

“…Therefore, many scholars have focused on some discrete population models [4,5,8,9,15,16,24,25,27]. But, at present, there is only a few research on discrete predator-prey system with non-monotonic functional response [5,15,27]. Chen et al proposed the following prey-predator discrete-time model with Holling-IV functional response by applying the Euler forward scheme,…”

“…By using the centre manifold theorem and bifurcation theory, they showed that the system (5) undergoes a flip bifurcation, a Hopf bifurcation and a saddle-node bifurcation [5]. Lu and Wang considered the following discrete semi-ratio-dependent predator-prey system with Holling-type IV functional response and time delay [15],…”

A dynamically consistent nonstandard finite difference scheme for a predator–prey model

“…Similarly, Liu and Xiao [5] presented complex dynamics for a discrete Lotka-Volterra system after implementation of Euler method. For a similar type of investigations related to predator-prey systems the interested reader is referred to [6][7][8][9][10][11][12][13][14][15][16][17]. All these studies reveal that the discrete predator-prey models with implementation of Euler approximation are dynamically inconsistent with their continuous counterparts.…”

Dynamics of a Diffusive Modified Leslie-Gower with Holling Type IV

“…Further, Aziz-Alaoui et al [8] investigated a predator-prey system with a modified version of Leslie-Gower and Holling type II schemes. Meanwhile, the traits of non-monotonic Holling type IV functional response have been clearly recognized and it is widely used to describe the process of predation with self-selection and the inhibitory effect of prey in the recent years [8][9][10][11][12]. Thus, we consider a diffusive modified Leslie-Gower with Holling type IV schemes, which can be described as following: 1 c is the maximum value of the per capita reduction of H due to P , 2 2 / c a measures the ration of prey to support one predator, 1 e is interpreted as the half-saturation constant, 2 e indicates the quality of the alternative that provides the environment, ∆ is the Laplacian operator, 1 d and 2 d are diffusion coefficient.…”